Analysis of the pion scalar form factor provides model independent values of f 0 ( 500 ) and f 0 ( 980 ) meson parameters

An existence of the scalar meson f0(500) is unambiguously confirmed by the pion scalar form factor analysis. The same is concerned also of the f0(980) scalar meson, though it is placed on the tail of the elastic region to be investigated in the analysis under consideration, therefore with less precise parameters values.


Introduction
In contrast to other S U(3) known multiplets of hadrons, the identification of the scalar mesons nonet [1] is long-standing puzzle.It is even more concerned of the lowest of them, the sigma-meson [2] [3], now to be called f 0 (500) resonance.
Therefore in this presentation first of all we demonstrate an existence of the f 0 (500) by a pion scalar form factor (FF) analysis.
With this aim we construct an explicit form of the pion scalar FF by using its phase representation and the best description of the S-wave iso-scalar ππ phase shift data by the parametrization in the absolute valued of the pion c.m. three-momentum q to be found starting from fully general considerations.

Pion scalar form factor phase representation
The pion scalar FF Γ π (t) is defined by the parametrization of the matrix element of the scalar quark density where t = (p 2 − p 1 ) 2 and m = 1 2 (m u + m d ).It posses all well known properties of the pion electromagnetic FF [4] [5].The analyticity and the asymptotic behavior allow to derive dispersion relations without subtractions or two subtractions etc., which in combination with the elastic unitarity condition (it produces the identity δ Γ ≡ δ 0 0 , where δ 0 0 is the S -wave isoscalar ππ phase shift and δ Γ is the phase of the pion scalar FF) give corresponding phase representations of Γ π (t) or etc., with arbitrary, however at t = 0 normalized, polynomial P n (t).Nevertheless, only a best describing existing data parametrization of δ 0 0 , obtained further fully from general considerations, can decide which of previous phase representations will be used to calculate an explicit form of the pion scalar FF.

Analysis of S 0 0 ππ phase shift data
By means of the elastic unitarity condition one can carry out analytic continuation of Γ π (t) through upper and lower boundaries of the two-pion cut on the II.Riemann sheet and to come to the same functional expression.The latter reveals that the two-pion branch point is a square-root type.As a result by application of the conformal mapping two-sheeted Riemann surface of Γ π (t) in t-variable is mapped into one absolute valued pion c.m. three-momentum q-plane and the elastic cut disappears.Neglecting all higher branch points, there are only poles and zeros of Γ π (t) in q-plane and Γ π (t) can be represented in the form of a rational function , which leads to the parametrization tan δ 0 0 (t) = A 1 q + A 3 q 3 + A 5 q 5 + A 7 q 7 + ... 1 + A 2 q 2 + A 4 q 4 + A 6 q 6 + ...
As the integrand there is even function of its argument, i.e. it is invariant under the transformation q → −q , the latter expression can be transformed into the following form q ln (1+A 2 q 2 +A 4 q 4 )+i(A 1 q +A 3 q 3 +A 5 q 5 ) (1+A 2 q 2 +A 4 q 4 )−i(A 1 q +A 3 q 3 +A 5 q 5 ) (q 2 + 1)(q 2 − q 2 ) dq in which the integral is suitable for calculation by means of the theory of residua.The calculations lead to the algebraic expression of the pion scalar FF Γ π (t) = P n (t) (q − q 1 ) (q + q 2 )(q + q 3 )(q + q 4 )(q + q 5 ) (i + q 2 )(i + q 3 )(i + q 4 )(i + q 5 ) (i − q 1 ) and its behavior at the elastic region is presented in Fig. 2. Now, investigating poles of the latter function, one finds that −q 3 and −q 2 on the second Riemann sheet in t-variable correspond to f 0 (500) and f 0 ( 980

Conclusions
By analysis of the pion scalar FF we have confirmed in a completely model independent way an existence of the f 0 (500) scalar meson resonance.With this aim we have started from the phase representation of the pion scalar FF with one subtraction and from a special parametrization of the S-wave iso-scalar ππ phase shift in the absolute value of the pion c.m. three-momentum q, found in general considerations.As a result pion scalar FF has been found in the form of a rational function and by investigation of its poles we have found f 0 (500), as well also f 0 (980) scalar meson resonance parameters.

Figure 1 .
Figure 1.Description of the S-wave iso-scalar ππ phase shift by the [5/4] Pad'e type approximation.